Graph theoretic proofs for some results on banded inverses of $M$-matrices

TL;DR

This paper uses graph theory to analyze conditions under which banded M-matrices have banded inverses, providing new characterizations and proofs for specific matrix structures.

Contribution

It introduces a graph theoretic characterization for positive off-diagonal inverse entries and offers alternative proofs for banded inverse conditions of M-matrices.

Findings

Graph theoretic characterization for positive off-diagonal inverse entries

Alternative proofs for tridiagonal M-matrix inverse conditions

Necessary conditions for pentadiagonal inverse of M-matrices

Abstract

This work concerns results on conditions guaranteeing that certain banded $M$-matrices have banded inverses. As a first goal, a graph theoretic characterization for an off-diagonal entry of the inverse of an $M$-matrix to be positive, is presented. This result, in turn, is used in providing alternative graph theoretic proofs of the following: (1) a characterization for a tridiagonal $M$-matrix to have a tridiagonal inverse. (2) a necessary condition for an $M$-matrix to have a pentadiagonal inverse. The results are illustrated by several numerical examples.